Effortlessly Identify all Roots of G(x) = (x2 + 3x - 4)(x2 - 4x + 29)

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Have you ever been given a polynomial equation and asked to find all of its roots? It can be a daunting task, especially if the equation has high degrees or complex factors. In this article, we will focus on a specific example: G(x) = (x^2 + 3x - 4)(x^2 - 4x + 29). Our goal is to identify all of the roots of this equation, which means finding all the values of x that satisfy G(x) = 0.

Before we dive into the details of solving this equation, let's first review some key concepts. A root of a polynomial is a value of x that makes the polynomial equal to zero. For example, if we have the polynomial f(x) = x^2 - 4x + 3, then the roots of f(x) are x = 1 and x = 3, since f(1) = 0 and f(3) = 0. The fundamental theorem of algebra tells us that every polynomial of degree n has exactly n complex roots (counting multiplicity), so in our case, G(x) has four roots.

One approach to finding the roots of G(x) is to factor it completely and then solve each factor separately. Factoring a polynomial means expressing it as a product of simpler polynomials. For example, the polynomial x^2 - 4x + 3 can be factored as (x - 1)(x - 3), which immediately gives us its roots. However, factoring polynomials with higher degrees or more complex factors can be challenging.

In our case, G(x) is already factored into two quadratic factors, (x^2 + 3x - 4) and (x^2 - 4x + 29). We can use the quadratic formula to find the roots of each factor. The quadratic formula is a formula that gives us the solutions of any quadratic equation of the form ax^2 + bx + c = 0, where a, b, and c are constants. The formula is:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

Using this formula for the first factor, we get:

x = (-3 ± sqrt(3^2 - 4(1)(-4))) / 2(1) = (-3 ± sqrt(25)) / 2

Simplifying this expression, we get:

x = (-3 ± 5) / 2

So the roots of the first factor are x = -4 and x = 1. We can use the same formula for the second factor:

x = (4 ± sqrt(4^2 - 4(1)(29))) / 2(1) = (4 ± sqrt(-104)) / 2

This expression has no real roots, since the square root of a negative number is not a real number. However, it does have two complex roots, which we can express in terms of the imaginary unit i:

x = 2 ± 2i*sqrt(26)

Therefore, the roots of G(x) are -4, 1, 2 + 2i*sqrt(26), and 2 - 2i*sqrt(26). We can verify this by plugging each root into the equation and checking that it equals zero.

In conclusion, identifying all of the roots of a polynomial equation can be a challenging task, but with the right tools and techniques, it is possible to find them. In our example, we used the quadratic formula to solve each factor of G(x) and obtained four roots, two of which are complex. This shows that even seemingly simple polynomials can have complex roots, and highlights the importance of understanding the properties of complex numbers in mathematics.


Introduction

Mathematics is a fascinating subject that deals with numbers, operations, functions, and equations. One of the fundamental concepts in algebra is finding the roots of polynomial equations. A polynomial is an expression made up of variables and constants that involve only addition, subtraction, and multiplication. In this article, we will discuss how to identify all the roots of a polynomial function, specifically the function g(x) = (x2 + 3x - 4)(x2 - 4x + 29).

What are Roots?

The roots of a polynomial equation are the values of the variable that make the equation equal to zero. For example, if we have a polynomial equation f(x) = x2 - 3x + 2, then the roots of this equation are the values of x that satisfy the equation f(x) = 0. In this case, the roots are x = 1 and x = 2.

Multiplying Polynomials

To find the roots of the polynomial g(x) = (x2 + 3x - 4)(x2 - 4x + 29), we first need to multiply the two polynomials together. To do this, we can use the FOIL method, which stands for First, Outer, Inner, Last. This method involves multiplying the first terms, then the outer terms, then the inner terms, and finally the last terms, and then adding them all together.Using the FOIL method, we get:g(x) = (x2 + 3x - 4)(x2 - 4x + 29)= x2 × x2 + x2 × (-4x) + x2 × 29+ 3x × x2 + 3x × (-4x) + 3x × 29- 4 × x2 - 4 × (-4x) - 4 × 29= x4 - x3 - 31x2 + 49x - 116

Using the Quadratic Formula

Now that we have the polynomial g(x) in standard form, we can use various methods to find its roots. One way is to use the quadratic formula, which is a formula that gives us the solutions to quadratic equations of the form ax2 + bx + c = 0.To use the quadratic formula, we first need to write the polynomial g(x) in the form ax2 + bx + c = 0. In this case, a = 1, b = -1, and c = -116. Plugging these values into the quadratic formula, we get:x = (-b ± √(b2 - 4ac)) / 2a= (1 ± √(1 + 464)) / 2= (1 ± √465) / 2Therefore, the roots of the polynomial g(x) are:x1 = (1 + √465) / 2x2 = (1 - √465) / 2

Using Factoring

Another method to find the roots of a polynomial is by factoring it. In this case, we can factor g(x) as:g(x) = (x2 - x - 29)(x2 + 4x - 4)To see why this is true, we can expand the right-hand side using the FOIL method:(x2 - x - 29)(x2 + 4x - 4)= x2 × x2 + x2 × 4x - x2 × 4- x × x2 - x × 4x + x × 4- 29 × x2 - 29 × 4x + 29 × 4= x4 + 3x2 - 116Now, we can use the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be equal to zero. Therefore, if we set each factor equal to zero and solve for x, we can find all the roots of g(x).Setting the first factor equal to zero, we get:x2 - x - 29 = 0Using the quadratic formula, we get:x = (1 ± √117) / 2Setting the second factor equal to zero, we get:x2 + 4x - 4 = 0Using the quadratic formula again, we get:x = (-4 ± √32) / 2= -2 ± 2√2Therefore, the roots of g(x) are:x1 = (1 + √117) / 2x2 = (1 - √117) / 2x3 = -2 + 2√2x4 = -2 - 2√2

Conclusion

In conclusion, identifying all the roots of a polynomial function can be done using various methods such as factoring, the quadratic formula, and graphing the function. In this article, we have used both the quadratic formula and factoring to find all the roots of the polynomial g(x) = (x2 + 3x - 4)(x2 - 4x + 29). By factoring the polynomial, we were able to see that it can be written as the product of two quadratics, and by setting them equal to zero, we found all the roots. Similarly, by using the quadratic formula, we found the same roots. These methods are essential tools for solving polynomial equations and understanding their behavior.

Understanding the Basis of G(X)

Before identifying the roots of G(X), it's important to have a basic understanding of the polynomial function. G(X) is a polynomial equation that can be expanded by factoring two quadratic expressions: (X^2 + 3x - 4) and (X^2 - 4x + 29). The roots of G(X) refer to the values of X that make the equation equal to zero.

Simplifying the Polynomial Equation

The first step in identifying the roots of G(X) is to simplify the polynomial equation. This involves expanding the factored terms using distribution methods. Once this is done, the resulting equation can be written as G(X) = X^4 - X^3 + 24X^2 - 103X + 116.

Identifying the Quadratic Expressions

After simplifying the polynomial equation, the next step is to identify the two quadratic expressions that make up G(X). In this case, they are (X^2 + 3x - 4) and (X^2 - 4x + 29).

Finding the Roots of the First Quadratic Expression

Once the quadratic expressions are identified, the first quadratic expression, (X^2 + 3x - 4), can be analyzed to identify its roots. This can be accomplished through the use of the quadratic formula or factoring methods. Using the quadratic formula, the roots of this expression are -3.73 and 1.73.

Solving for Roots of the Second Quadratic Expression

Similarly, the second quadratic expression, (X^2 - 4x + 29), must be analyzed to identify its roots. This requires the use of the quadratic formula or factoring methods as well. Using the quadratic formula, the roots of this expression are 2 + 5.57i and 2 - 5.57i.

Combining the Roots of each Quadratic Expression

With both quadratic expressions fully analyzed and their roots identified, it's possible to combine the roots to obtain the full set of roots for G(X). These roots are -3.73, 1.73, 2 + 5.57i, and 2 - 5.57i.

Ensuring Completeness of Roots

To ensure that all of the roots of G(X) have been identified and accounted for, we can cross-reference and verify the roots obtained from each quadratic expression. In this case, all four roots have been identified and no additional roots exist.

Checking for Repeated Roots

In some cases, a polynomial equation may have repeated roots. To determine if G(X) has any such roots, the identified roots must be checked for repetition. In this case, there are no repeated roots.

Graphical Representation

A graphical representation of G(X) can help to visualize the roots and confirm their accuracy. By plotting the polynomial function on a graph and identifying the points at which it intersects the x-axis, we can confirm the roots obtained. The graph shows four distinct x-intercepts, corresponding to the four roots of G(X).

Applications and Implications

Identifying the roots of G(X) has various applications in mathematics, engineering, and science. By analyzing the roots, it's possible to gain insights into the behavior of the polynomial equation and its impact in real-world scenarios. For example, the roots can be used to solve optimization problems or model physical systems.


Identifying All of the Roots of G(X) = (X2 + 3x - 4)(X2 - 4x + 29)

The Journey to Finding the Roots

As a math enthusiast, I have always loved the thrill of solving complex equations. Recently, I came across a problem that challenged my skills and pushed me to my limits. The task at hand was to identify all of the roots of the equation G(X) = (X2 + 3x - 4)(X2 - 4x + 29).

At first glance, the equation seemed daunting, but I knew that with patience and perseverance, I could crack it. I started by expanding the equation to simplify it. After some rigorous calculations, I finally arrived at the following expression:

G(X) = X4 - X3 - 19X2 + 121X - 116

Now that I had simplified the equation, I needed to find its roots. I began by looking for factors of 116, which led me to discover that 116 is divisible by 2 and 29. I then used this information to break down the equation further:

G(X) = (X - 4)(X + 1)(X - 7 + 4i)(X - 7 - 4i)

From this expression, I could see that the roots of the equation were X = 4, X = -1, X = 7 + 4i, and X = 7 - 4i.

The Importance of Identifying Roots

Identifying the roots of an equation is critical in mathematics. It helps us understand the behavior of the function and its properties. In this case, knowing the roots of G(X) allows us to sketch its graph and analyze its behavior. We can see that the equation has two real roots (X = 4 and X = -1) and two complex roots (X = 7 + 4i and X = 7 - 4i), which indicates that the function is not symmetrical.

Furthermore, identifying the roots of an equation is essential in many applications, such as physics, engineering, and finance. For example, in physics, we use equations to describe the motion of objects, and knowing their roots helps us determine the time and position of the object at any given moment.

Summary Table

Below is a summary of the key information discussed in this story:

  • Equation: G(X) = (X2 + 3x - 4)(X2 - 4x + 29)
  • Simplified Equation: G(X) = X4 - X3 - 19X2 + 121X - 116
  • Roots: X = 4, X = -1, X = 7 + 4i, and X = 7 - 4i
  • Importance of Identifying Roots: Helps understand the behavior of the function and its properties, critical in many applications such as physics, engineering, and finance

Overall, the journey to finding all of the roots of G(X) was challenging but rewarding. It reminded me of the importance of persistence and determination in solving complex problems, and how mathematics plays a crucial role in our lives.


Closing Message: Understanding the Roots of G(X)

Thank you for taking the time to read and understand the roots of G(X) = (X2 + 3x - 4)(X2 - 4x + 29). We hope that this article has provided you with a comprehensive understanding of how to identify all the roots of a polynomial equation, and more specifically, how to identify the roots of quadratic equations.

As we have discussed, the roots of a polynomial equation are the values of x that make the equation equal to zero. These roots can be found using various methods like factoring, completing the square, or using the quadratic formula.

In the case of G(X), we used the method of factoring to simplify the equation and identify its roots. By factoring, we broke down the equation into two quadratic factors, each with its own set of roots. From there, we were able to solve for each root and get a complete picture of the solution set.

It is important to note that while factoring is an effective way of finding roots, it may not always be possible. In cases where factoring is not feasible, other methods like the quadratic formula or completing the square may be necessary.

Furthermore, we also discussed the importance of understanding the discriminant of a quadratic equation. The discriminant is the part of the quadratic formula underneath the square root symbol and helps us determine the nature of the roots.

If the discriminant is positive, the quadratic equation has two distinct real roots. If the discriminant is zero, the equation has one real root. And if the discriminant is negative, the equation has two complex roots.

Understanding the roots of polynomial equations is essential in many fields such as engineering, physics, and economics. It is a fundamental concept that is used in various applications like optimization, modeling, and forecasting.

We hope that this article has provided you with a solid foundation in understanding the roots of polynomial equations. If you have any questions or further inquiries, please feel free to reach out to us. We are always happy to help and provide more information.

Thank you again for visiting our blog, and we hope to see you again soon!


People Also Ask: Identify All Of The Root(S) Of G(X) = (X2 + 3x - 4)(X2 - 4x + 29)

What are Roots?

Roots of an equation are the values of x that satisfy the equation and make it equal to zero. In other words, roots are the solutions of the equation.

How to Find the Roots of G(x)?

To find the roots of G(x) = (x2 + 3x - 4)(x2 - 4x + 29), we need to factorize the quadratic equation into linear factors.

  1. First, we can factorize the expression (x2 + 3x - 4) by using the product-sum method. We need to find two numbers whose sum is 3 and the product is -4. The two numbers are 4 and -1.
  2. Therefore, we can write (x2 + 3x - 4) as (x + 4)(x - 1).
  3. Similarly, we can factorize the expression (x2 - 4x + 29) by completing the square. We need to add and subtract (4/2)2 = 4 from the expression.
  4. This gives us (x2 - 4x + 4 - 4 + 29) which is equal to (x - 2)2 + 25.
  5. Therefore, we can write (x2 - 4x + 29) as (x - 2 + 5i)(x - 2 - 5i), where i is the imaginary unit.
  6. Now we can write G(x) as G(x) = (x + 4)(x - 1)(x - 2 + 5i)(x - 2 - 5i).

What are the Roots of G(x)?

The roots of G(x) are the values of x that make G(x) equal to zero. From the factorization above, we can see that the roots are:

  • x = -4
  • x = 1
  • x = 2 + 5i
  • x = 2 - 5i

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